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Exponents with negative bases

Learn to what we know about negative numbers to determine how negative bases with exponents are affected and what patterns develop. Also learn how order of operations affect the pattern. Created by Sal Khan.

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  • aqualine ultimate style avatar for user jrgas
    Is there a real life situation where an exponent with a negative base (-x^3) would be used? Could you give me a word problem using it?
    (23 votes)
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    • blobby green style avatar for user Larrywisecarver
      a real-life situation where you might use a positive exponent with a negative base. Suppose you're a scientist studying the behavior of a certain type of bacteria. You notice that every hour, the number of bacteria decreases by a factor of 5. After 4 hours, how many times less bacteria will there be compared to the original amount? In this case, you can use the expression -5^4 to calculate the change in the number of bacteria. The base -5 represents the factor by which the number of bacteria decreases every hour, and the exponent 4 represents the number of hours. According to the order of operations, you should calculate 5^4 first, then multiply the result by -1. So, -5^4 = -1*5*5*5*5 = -625. This means that after 4 hours, there will be 625 times less bacteria than the original amount.
      (9 votes)
  • winston default style avatar for user Anderson Le
    How come I'm practicing 8th grade math and it states on one of the problems that negative 7 squared is negative 49. I thought it was positive. Apparently, the problem stated in the hints that the negative sign is not part of the base of the exponent and therefore, is a negative number. I did not quite understand this concept.
    (12 votes)
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  • duskpin ultimate style avatar for user zachbroyles64
    at I get the exponent but I don't get negative.
    (4 votes)
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  • duskpin seedling style avatar for user Isabella
    Why do we need negative exponents? If so how do we use negative exponents?
    (3 votes)
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  • mr pants teal style avatar for user ThatSmartCookieAbby
    i have been watching this video and trying to solve problems about this topic, but for some reason i keep getting them wrong. i don't understand the negative base. can anyone help me please?
    (4 votes)
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    • piceratops ultimate style avatar for user Ryan
      The problem you are having is that multiplying a negative number with a negative number is a positive. So, when you have a negative base, it will always be positive. For example:

      (-8)^2
      To solve this, all you have to do is multiply -8 by -8, or -8 x -8. As you can see, multiplying a negative number by a negative equals a positive.
      Answer: 64

      Hope this helps you!
      (3 votes)
  • starky ultimate style avatar for user Taders09
    my mind is blown, I mean like who would think that if you do -4 to the second power and you do (-4)to the second power you get different answers!
    (5 votes)
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  • cacteye purple style avatar for user Francie
    (-2) ^2

    So for this, it would just be - 2 ✕ - 2 ? Just checking!

    Thanks!
    (5 votes)
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  • mr pink red style avatar for user MANOJ SHARMA
    A solar powered water system has a solar panel containg 8 rows of moduels . How many modules are there in each row if there are 8 raise to the power 8 modules in all
    (3 votes)
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    • piceratops ultimate style avatar for user fr33d0g
      Ok, lets say you had 8^2 , that would give you 64 modules right. but if you had 8 rows you would divide that number by 8, correct? That would give you 8 modules in 8 rows. Remember multiplication and division are the opposite of each other. So if you had a total of 8^8 modules in all, and you divided that number by 8 it would give you 8^7. If you divided 8^7 by 8 it would give you 8^6, and so on and so on. So if you have 8^8 modules in all, and you want to know how many in each row, and you have 8 rows you would divide 8^8 by 8 which would give you 8^7 which equals 2,097,152. Hope this helps :)
      (2 votes)
  • aqualine ultimate style avatar for user bhaikabirsingh
    On a calculator -3^2 equals -9 but (-3)^2 equals 9. Why is that?
    (1 vote)
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    • primosaur seed style avatar for user Ian Pulizzotto
      Without parentheses, the exponent is applied before the negative sign.

      However, if the negative number is in parentheses, then the exponent is applied to the entire negative number.

      In the expression -3^2, 3^2 is calculated to give 3*3 = 9, then the negative sign is attached to give a final answer of -9. It helps to think of attaching the negative sign as multiplying by -1, and to recall that exponents are performed before multiplication according to PEMDAS.

      However, in the expression (-3)^2, the exponent 2 is applied to the entire number -3, so (-3)^2 = (-3)*(-3) = 9.

      Have a blessed, wonderful day!
      (6 votes)
  • aqualine ultimate style avatar for user Grace Miedziak
    If I see -4^2 with no parenthesis, can I safely assume that it would be positive 16?
    (2 votes)
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Video transcript

Let's see if we can apply what we know about negative numbers, and what we know about exponents to apply exponents to negative numbers. So let's first think about – Let's say we have -3. Let's first think about what it means to raise it to the 1st power. Well that literally means just taking a -3. And there's nothing left to multiply it with. So this is just going to be equal to -3. Now what happens if you were take a -3, and we were to raise it to the 2nd power? Well that's equivalent to taking 2 -3's, so a -3 and a -3, and then multiplying them together. What's that going to be? Well a negative times a negative is a positive. So that is going to be positive 9. Let me write this. It's going to be positive 9. Well, let's keep going. Let's see if there is some type of pattern here. Let's take -3 and raise it to the 3rd power. What is this going to be equal to? Well, we're going to take 3 -3's, [WRITING] – and we're going to multiply them together. So we're going to multiply them together. -3 × -3, we already figured out is positive 9. But positive 9 × -3, well that's that's -27. And so you might notice a pattern here. Whenever we raised raised a negative base to an exponent, if we raise it to an odd exponent, we are going to get a negative value. And that's because when you multiply negative numbers an even number of times, a negative number times a negative number is a positive. But then you have one more negative number to multiply the result by – which makes it negative. And if you take a negative base, and you raise it to an even power, that's because if you multiply a negative times a negative, you're going to get a positive. And so when you do it an even number of times, doing it a multiple-of-two number of times. So the negatives and the negatives all cancel out, I guess you could say. Or when you take the product of the two negatives, you keep getting positives. So this right over here is going to give you a positive value. So there's really nothing new about taking powers of negative numbers. It's really the same idea. And you just really have to remember that a negative times a negative is a positive. And a negative times a positive is a negative, which we already learned from multiplying negative numbers. Now there's one other thing that I want to clarify – because sometimes there might be ambiguity if someone writes this. Let's say someone writes that. And I encourage you to actually pause the video and think about with this right over here would evaluate to. And, if you given a go at that, think about whether this should mean something different then that. Well this one can be a little bit and big ambiguous and if people are strict about order of operations, you should really be thinking about the exponent before you multiply by this -1. You could this is implicitly saying -1 × 2^3. So many times, this will usually be interpreted as negative 2 to the third power, which is equal to -8, while this is going to be interpreted as -2 to the third power. Now that also is equal to -8. You might say well what's what's the big deal here? Well what if this was what if these were even exponents. So what if someone had give myself some more space here. What if someone had these to express its -4 or a -4 squared or -4 squared. This one clearly evaluates to 16 – positive 16. It's a negative 4 times a *4. This one could be interpreted as is. Especially if you look at order of operations, and you do your exponent first, this would be interpreted as -4 times 4, which would be -16. So it's really important to think about this properly. And if you want to write the number negative if you want the base to be negative 4, put parentheses around it and then write the exponent.